High-level convexity for products of squared Euclidean distance functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913174895198208 |
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| author | Micu, Tudor Pintea, Cornel Ţurcaş, George C. |
| author_facet | Micu, Tudor Pintea, Cornel Ţurcaş, George C. |
| contents | We study smooth functions on Euclidean space whose Hessian is positive definite outside a bounded set, with emphasis on products of squared distance functions. More precisely, we first prove a simple convexity principle: if the superlevel region $f^{-1}([c,\infty))$ is contained in the Hessian-positive region of $f$, then the sublevel set $\{f\le c\}$ is convex. We apply this to finite products $F_P(x)=\prod_{p\in P}\|x-p\|^2$, proving that their Hessian-positive complements are bounded. For the two-centre product $F_{p,q}(x)=\|x-p\|^2\|x-q\|^2$ in dimension $n\ge2$, we compute the Hessian-positive region and the exact value \[
h_{max}(F_{p,q})=\frac{\|p-q\|^4}{4}. \] This value is sharp for convexity of sublevel sets in the following sense: we prove convexity above it and nonconvexity below it. This also gives the exact convexity and quasiconvexity truncation levels for the two-centre model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_00316 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High-level convexity for products of squared Euclidean distance functions Micu, Tudor Pintea, Cornel Ţurcaş, George C. Classical Analysis and ODEs Differential Geometry 26B25 We study smooth functions on Euclidean space whose Hessian is positive definite outside a bounded set, with emphasis on products of squared distance functions. More precisely, we first prove a simple convexity principle: if the superlevel region $f^{-1}([c,\infty))$ is contained in the Hessian-positive region of $f$, then the sublevel set $\{f\le c\}$ is convex. We apply this to finite products $F_P(x)=\prod_{p\in P}\|x-p\|^2$, proving that their Hessian-positive complements are bounded. For the two-centre product $F_{p,q}(x)=\|x-p\|^2\|x-q\|^2$ in dimension $n\ge2$, we compute the Hessian-positive region and the exact value \[ h_{max}(F_{p,q})=\frac{\|p-q\|^4}{4}. \] This value is sharp for convexity of sublevel sets in the following sense: we prove convexity above it and nonconvexity below it. This also gives the exact convexity and quasiconvexity truncation levels for the two-centre model. |
| title | High-level convexity for products of squared Euclidean distance functions |
| topic | Classical Analysis and ODEs Differential Geometry 26B25 |
| url | https://arxiv.org/abs/2606.00316 |